5.8: Exercises-1- Skills
- Page ID
- 37761
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- Chance and Brianna buy a pizza for $10 that is half pepperoni and half veggie. They cut the pizza into 8 slices.
If Chance likes veggie three times as much as pepperoni, what is the value of a slice that is half pepperoni, half veggie?
- Ahmed and Tiana buy a cake for $14 that is half chocolate and half vanilla. They cut the cake into 8 slices.
If Ahmed likes chocolate four times as much as vanilla, what is the value of a slice that is half chocolate, half vanilla?
- Erin, Catherine, and Shannon are dividing a large bag of candy. They randomly split the bag into three bowls. The values of the entire bag and each of the three bowls in the eyes of each of the players are shown below. For each player, identify which bowls they value as a fair share.
\(\begin{array}{|c|l|l|l|l|}
\hline & {\textbf { Whole Bag }} & {\textbf { Bowl 1 }} & {\textbf { Bowl 2 }} & {\textbf { Bow1 3 }} \\
\hline \textbf { Erin } & \$ 5 & \$ 2.75 & \$ 1.25 & \$ 1.00 \\
\hline \textbf { Catherine } & \$ 4 & \$ 0.75 & \$ 2.50 & \$ 0.75 \\
\hline \textbf { Shannon } & \$ 8 & \$ 1.75 & \$ 2.25 & \$ 4.00 \\
\hline
\end{array}\)
- Jenna, Tatiana, and Nina are dividing a large bag of candy. They randomly split the bag into three bowls. The values of the entire bag and each of the three bowls in the eyes of each of the players are shown below. For each player, identify which bowls they value as a fair share.
\(\begin{array}{|c|l|l|l|l|}
\hline & \textbf { Whole Bag } & \textbf { Bowl 1 } & \textbf { Bowl 2 } & \textbf { Bow1 3 } \\
\hline \textbf { Jenna } & \$ 8 & \$ 4.50 & \$ 0.75 & \$ 2.75 \\
\hline \textbf { Tatiana } & \$ 4 & \$ 1.00 & \$ 1.00 & \$ 2.00 \\
\hline \textbf { Nina } & \$ 6 & \$ 1.75 & \$ 2.50 & \$ 1.75 \\
\hline
\end{array}\)
- Dustin and Kendra want to split a bag of fun-sized candy, and decide to use the divider-chooser method. The bag contains 100 Snickers, 100 Milky Ways, and 100 Reese's, which Dustin values at $1, $5, and $2 respectively. (This means Dustin values the 100 Snickers together at $1, or $0.01 for 1 Snickers).
If Kendra is the divider, and in one half puts: 25 Snickers, 20 Milky Ways, and 60 Reese's
- What is the value of this half in Dustin's eyes?
- Does Dustin consider this a fair share?
- If Dustin was a divider, find a possible division that is consistent with his value system.
- Dustin and Kendra want to split a bag of fun-sized candy, and decide to use the divider-chooser method. The bag contains 100 Snickers, 100 Milky Ways, and 100 Reese's, which Dustin values at $1, $3, and $5 respectively. (This means Dustin values the 100 Snickers together at $1, or $0.01 for 1 Snickers).
If Kendra is the divider, and in one half puts: 30 Snickers, 40 Milky Ways, and 66 Reese's
- What is the value of this half in Dustin's eyes?
- Does Dustin consider this a fair share?
- If Dustin was a divider, find a possible division that is consistent with his value system.
- Maggie, Meredith, Holly, and Zoe are dividing a piece of land using the lone-divider method. The values of the four pieces of land in the eyes of the each player are shown below.
\(\begin{array}{|c|l|l|l|l|}
\hline & \textbf { Piece 1 } & \textbf { Piece 2 } & \textbf { Piece 3 } & \textbf { Piece 4 } \\
\hline \textbf { Maggie } & 21 \% & 27 \% & 32 \% & 20 \% \\
\hline \textbf { Meredith } & 27 \% & 29 \% & 22 \% & 22 \% \\
\hline \textbf { Holly } & 23 \% & 14 \% & 41 \% & 22 \% \\
\hline \textbf { Zoe } & 25 \% & 25 \% & 25 \% & 25 \% \\
\hline
\end{array}\)
- Who was the divider?
- If playing honestly, what will each player’s declaration be?
- Find the final division.
- Cody, Justin, Ahmed, and Mark are going to share a vacation property. The year will be divided into 4 time slots using the lone-divider method. The values of each time slot in the eyes of the each player are shown below.
\(\begin{array}{|c|l|l|l|l|}
\hline & \textbf { Time 1 } & \textbf { Time 2 } & \textbf { Time 3 } & \textbf { Time 4 } \\
\hline \textbf { Cody } & 10 \% & 35 \% & 34 \% & 21 \% \\
\hline \textbf { Justin } & 25 \% & 25 \% & 25 \% & 25 \% \\
\hline \textbf { Ahmed } & 19 \% & 24 \% & 30 \% & 27 \% \\
\hline \textbf { Mark } & 23 \% & 31 \% & 22 \% & 24 \% \\
\hline
\end{array}\)
- Who was the divider?
- If playing honestly, what will each player’s declaration be?
- Find the final division.
9. A 6-foot sub valued at $30 is divided among five players \((P_{1}, P_{2}, P_{3}, P_{4}, P_{5})\) using the last-diminisher method. The players play in a fixed order, with P1 first, P2 second, and so on. In round 1, P1 makes the first cut and makes a claim on a piece. For each of the remaining players, the value of the current claimed piece at the time it is their turn is given in the following table:
\(\begin{array}{|l|l|l|l|l|}
\hline & \mathbf{P}_{2} & \mathbf{P}_{3} & \mathbf{P}_{4} & \mathbf{P}_{5} \\
\hline \text { Value of the current claimed piece } & \$ 6.00 & \$ 8.00 & \$ 7.00 & \$ 6.50 \\
\hline
\end{array}\)
- Which player gets his or her share at the end of round 1?
- What is the value of the share to the player receiving it?
- How would your answer change if the values were:
\(\begin{array}{|l|l|l|l|l|}
\hline & \mathbf{P}_{2} & \mathbf{P}_{3} & \mathbf{P}_{4} & \mathbf{P}_{5} \\
\hline \text { Value of the current claimed piece } & \$ 6.00 & \$ 8.00 & \$ 7.00 & \$ 4.50 \\
\hline
\end{array}\)
10. A huge collection of low-value baseball cards appraised at $100 is being divided by 5 kids \((P_{1}, P_{2}, P_{3}, P_{4}, P_{5})\) using the last-diminisher method. The players play in a fixed order, with P1 first, P2 second, and so on. In round 1, P1 makes the first selection and makes a claim on a pile of cards. For each of the remaining players, the value of the current pile of cards at the time it is their turn is given in the following table:
\(\begin{array}{|l|l|l|l|l|}
\hline & \mathbf{P}_{2} & \mathbf{P}_{3} & \mathbf{P}_{4} & \mathbf{P}_{5} \\
\hline \text { Value of the current pile of cards } & \$ 15.00 & \$ 22.00 & \$ 18.00 & \$ 19.00 \\
\hline
\end{array}\)
- Which player gets his or her share at the end of round 1?
- What is the value of the share to the player receiving it?
- How would your answer change if the values were:
\(\begin{array}{|l|l|l|l|l|}
\hline & \mathbf{P}_{2} & \mathbf{P}_{3} & \mathbf{P}_{4} & \mathbf{P}_{5} \\
\hline \text { Value of the current pile of cards } & \$ 15.00 & \$ 22.00 & \$ 18.00 & \$ 21.00 \\
\hline
\end{array}\)
- Four heirs (A, B, C, and D) must fairly divide an estate consisting of two items - a desk and a vanity - using the method of sealed bids. The players' bids (in dollars) are:
\(\begin{array}{|c|c|c|c|c|}
\hline & \mathbf{A} & \mathbf{B} & \mathbf{C} & \mathbf{D} \\
\hline \textbf { Desk } & 320 & 240 & 300 & 260 \\
\hline \textbf { Vanity } & 220 & 140 & 200 & 180 \\
\hline
\end{array}\)
- What is A’s fair share?
- Find the initial allocation.
- Find the final allocation.
- Three heirs (A, B, C) must fairly divide an estate consisting of three items - a house, a car, and a coin collection - using the method of sealed bids. The players' bids (in dollars) are:
\(\begin{array}{|c|r|r|r|}
\hline & \mathbf{A} & \mathbf{B} & \mathbf{C} \\
\hline \textbf { House } & 180,000 & 210,000 & 220,000 \\
\hline \textbf { Car } & 12,000 & 10,000 & 8,000 \\
\hline \textbf { Coins } & 3,000 & 6,000 & 2,000 \\
\hline
\end{array}\)
- What is A’s fair share?
- Find the initial allocation.
- Find the final allocation.
- As part of an inheritance, four children, Abby, Ben and Carla, are dividing four vehicles using Sealed Bids. Their bids (in thousands of dollars) for each item is shown below. Find the final allocation.
\(\begin{array}{|c|r|r|r|}
\hline & \textbf { Abby } & \textbf { Ben } & \textbf { Carla } \\
\hline \textbf { Motorcycle } & 10 & 9 & 8 \\
\hline \textbf { Car } & 10 & 11 & 9 \\
\hline \textbf { Tractor } & 4 & 1 & 2 \\
\hline \textbf { Boat } & 7 & 6 & 4 \\
\hline
\end{array}\)
- As part of an inheritance, four children, Abby, Ben, Carla, and Dan, are dividing four vehicles using Sealed Bids. Their bids (in thousands of dollars) for each item is shown below. Find the final allocation.
\(\begin{array}{|c|r|r|r|r|}
\hline & \textbf { Abby } & \textbf { Ben } & \textbf { Carla } & \textbf { Dan } \\
\hline \textbf { Motorcycle } & 6 & 7 & 11 & 8 \\
\hline \textbf { Car } & 8 & 13 & 10 & 11 \\
\hline \textbf { Tractor } & 3 & 1 & 5 & 4 \\
\hline \textbf { Boat } & 7 & 6 & 3 & 8 \\
\hline
\end{array}\)
- After living together for a year, Sasha and Megan have decided to go their separate ways. They have several items they bought together to divide, as well as some moving-out chores. The values each place are shown below. Find the final allocation.
\(\begin{array}{|c|r|r|}
\hline & \textbf { Sasha } & \textbf { Megan } \\
\hline \textbf { Couch } & 120 & 80 \\
\hline \textbf { TV } & 200 & 250 \\
\hline \textbf { Stereo } & 40 & 50 \\
\hline \textbf { Detail cleaning } & -40 & -60 \\
\hline \textbf { Cleaning carpets } & -50 & -40 \\
\hline
\end{array}\)
- After living together for a year, Emily, Kayla, and Kendra have decided to go their separate ways. They have several items they bought together to divide, as well as some moving-out chores. The values each place are shown below. Find the final allocation.
\(\begin{array}{|c|r|r|r|}
\hline & \textbf { Emily } & \textbf { Kayla } & \textbf { Kendra } \\
\hline \textbf { Dishes } & 20 & 30 & 40 \\
\hline \textbf { Vacuum cleaner } & 100 & 120 & 80 \\
\hline \textbf { Dining table } & 100 & 80 & 130 \\
\hline \textbf { Detail cleaning } & -70 & -40 & -50 \\
\hline \textbf { Cleaning carpets } & -30 & -60 & -50 \\
\hline
\end{array}\)